Download Fredericks transitions induced by light fields

Fredericks transitions induced by light fields
B. Ya. Zel'dovich, N. V. Tabiryan, and Yu. S. Chilingaryan
Erewn State University
(Submitted2 December 1980)
Zh. Eksp. Teor. Fiz. 8 1 , 7 2 4 3 (July 1981)
The reorientation of the director of a nematic liquid cwstal induced by the field of a light wave is considered.
An oblique (with respect to the director) extraordinary wave of low intensity yields the predicted and
previously observed giant optical nonlinearity in a nematic liquid crystal. For normal incidence of the light
wave on the cuvette with a homeotropic orientation of the nematic liquid crystal, the reorientation appears
only at light intensities above a certain threshold, and the process itself is similar to the Fredericks transition.
The spatial distribution of the director direction is calculated for intensities above and below threshold.
Hysteresis of the Fredericks transition in a light field, which has no analog in the case of static fields, is
predicted.
PACS numbers: 61.30.Gd, 6 4 . 7 0 . E ~
1. INTRODUCTION
The nonlinear optics of liquid crystals has recently
received a great deal of attention (see the review1).
I n addition to the quite interesting quadratic nonlinearities (generation of the second harmonic2), recently
there have been prediction^^-^ and experimental disc o v e r i e of
~ ~giant
~ ~ cubic optical nonlinearities of liquid
crystals (nematic, cholesteric, and smectic), which
a r e due to the reorientation of the director by light
. ~ ~confields. Succeeding experimental s t ~ d i e s ' ~have
firmed the presence of giant optical nonlinearities for
a number of specific nematic liquid crystals (NLC) and
experimental geometries. In particular, a nonlinear
interaction was found" between a normally incident light
wave and a homeotropically oriented cell of the liquid
crystal (OCBP),and had a characteristic threshold dependence on the intensity of the l a s e r beam. This effect
was explained qualitatively in Ref. 11 on the basis of an
analogy with the Fredericks transition in the field of a
light wave.
In the present study we construct a quantitative theory
of the Fredricks transition in the field of a light wave.
In contrast to the simplest model of the Fredericks transition in static fields (see Refs. 12 and 13), here we
take into account the following two facts: 1) the amplitude for the deformation of the director above threshold must be found taking into account the distortion of
the longitudinal profile of the light wave itself a s it
propagates in an inhomogeneous anisotropic medium;
2) if the transverse dimension of the beam is significantly smaller than the cuvette thickness, the threshold
intensity itself depends on the transverse distribution of
the intensity in the beam. In addition, the theory predicts that for certain liquid crystals the Fredericks
transition is accompanied by hysterisis.
2. THE SYSTEM OF BASIC EQUATIONS
We shall take the free energy per unit volume of an
NLC i n the presence of a light field of complex amplitude
E to be of the form
F [ergicm'l ='/2Kl, ( d i n)2+112K?2(n
~
rot n)'
L'11K13[n rotnIZ-t.(nE) (nE')/lGrr-~_IE 1 2 / 1 6 ~ .
32
Sov. Phys. JETP 5 4 1 ), July 1981
Here K,, a r e the Frank constants, n i s a unit vector in
the direction of the director (nand -n will be assumed
equivalent), and the permittivity tensor of an NLC a t the
frequency of the light field w is
&,*=&,6,*+(E,,-E,.)
n,n~.
In addition, we have used in (1) the notation c , = E,,- cl.
The complex amplitude of a quasimonochromatic field
E ( r , t) is determined in a self-consistent manner from
the solution of Maxwell's equations with
E,r
(r, t) =8~6t~+&cnt(r,
t)
t).
However, we emphasize that to obtain the variational
equations for the director n(r, t) it is necessary to a s sume that the amplitude of the electric field E ( r , t) is
fixed. We shall take the density of the dissipative function R (in erg/cm3 s e c ) to be
(2)
R='/,I~(nn).
T h e variational equations for n(r ,t) have the form
--6F
6ni
6R
a
6F
=Sn,-ax, s(anilaxA)
6ni '
(3)
is an undetermined Lagrange multiplier that
where
ensures that the condition I nl = 1 is satisfied.
3. THE FREDERICKS TRANSITION IN BROAD LIGHT
BEAMS
Let us consider a homeotropically oriented NLC cell
occupying a layer 0 a z c L. We shall assume that the
light field has nonzero components Ex and E,, while E ,
= 0. The unperturbed direction of the director in n o =e,.
We shall also assume that the perturbation of the director does not move it out of the xz plane. Then
n(r) =e, cos q+e. sin cp,
where cp =cp(r). If the transverse dimensions of the
beam a, in the xy plane a r e much larger than the cuvette
thickness L and the beam itself is almost a plane wave,
the field components Ex and E, can be assumed to depend only on z ; more precisely,
E=[e,E,(z) +e,E, (z) ] esp ( i o s x l c + i k , z ) .
(4)
Here s =sins and
ff
is the angle of incidence of the
O 1982 American Institute of Physics
32
cp(z,
nkz
t)=C sin-exp
L
rht,
cp,
I
F r o m (7) we find that f o r
FIG. 1. Unperturbed director vector no perpendicular to the
cell walls. The wave vector k'of the light field makes an angle
with the normal to the cell walls.
light wave on t h e cell in a i r ( s e e Fig. 1). Under these
conditions the slope of the d i r e c t o r rp a l s o depends only
on 2 . Now the variational equations (3) take the f o r m
ascp
( a , ,sinzcp+K,, cos2cp)-
a z2 -(K.-K,,)sin
cp cos cp
acp
(x)
Let u s f i r s t give a qualitative picture of the F r e d e r i c k s
transition. When a plane light wave falls a t normal incidence on a homeotropic cell the e l e c t r i c field of the
wave is exactly perpendicular to the director. A t a positive value of E, it would be energetically favorable t o
orient the d i r e c t o r in the direction of the field. However,
this is prevented by the homeotropic orientation of the
d i r e c t o r by the curvette walls. F u r t h e r m o r e , in
the f i r s t approximation in the light intensity t h e
orientational effect of the field on the unperturbed
d i r e c t o r is absent, in other words, for E =e,Ex the
function cp(z) = 0 is a n exact solution to equation (5).
Above a certain threshold value of the intensity the
solution (p(z) = 0 will no longer be stable. T o determine
this threshold it is convenient to consider the linearized [that i s , a t s m a l l (p(z)] equation (5). H e r e we m u s t
take into account the fact that f r o m the equation d i v D = O
we have
T h e r e f o r e , to t e r m s linear in (p
and equation (5) takes the f o r m
a perturbation of the f o r m s i n ( n z / l ) begins to grow
exponentially.
sin(nz/~)
T h e steady-state value of theamplitude cp =, , ,pc
in the r e g i m e above threshold m u s t now be xetermined
with allowance for the nonlinear t e r m s in equation (5).
I t is v e r y important that in this approximation we m u s t
include a solution consistent with (p(z) to Maxwell's equations in a n inhomogeneous anisotropic medium. F o r a
wave of the f o r m (4) these solutions c a n be obtained in
the geometrical optics approximation ( s e e Refs. 14 and
15):
E,(z) = A ( E , , - s ~ ) ' ~ ~ ' ~ ( ~ ' ,
E..
E,(z)=-A
(e.,--sa)"+s(eIle,)'"
e,, (E,,-s')"
e i.rcz,.
T h e advance of the optical phase $ ( z ) is equal t o
o
*(:)=-J
-
" - s e . , ( ~ ' ) + I e , ~ ~ ~ ( e . , ( z ' ) -I"'s ~ )d Z ,
e,.(z')
0
A s a l r e a d y noted, s =sins, where a is the angle of
incidence in a i r , and t h e E,, a r e
s,,=e, sin cp ( z ) cos cp ( z ) .
E,,=E,+E. cosZq ( z ) ,
(9)
T h e constant A c a n b e e x p r e s s e d in t e r m s of the power
flux density; namely the z component of the Poynting
vector is
P , = C ( E ~ ~ lz/8n
~ ~ ) ' ~ ~ A
and is independent of
2.
I n the c a s e of exactly n o r m a l incidence (s =0) equation (5), taking into account (8) and (9), has the followwe have E , = - E , ( ~ E ~ / E ~ , ing f o r m i n the stationary c a s e :
d Z 9-(K,,-K,,)sin cp cos cp
( K i tsinzcp+K~scoszcp)dz2
e.elIe,lAI2 sincpcos cp
8%(e,+e. cosZcp)" = O.
+
Equation (6) is analogous to the linearized equation for
the behavior of the d i r e c t o r n e a r the F r e d e r i c k s
transition in a s t a t i c field E ,,,. = e x E,,, (cf. Ref. 13,
section 4.2). T h e r e a r e two differences, however.
of the s t a t i c c a s e in (6) we
F i r s t , instead of the
have I E, 12/2-the mean s q u a r e of the amplitude of the
light field. Secondly, in a s t a t i c field E,,,the condition
d i v D = O has no effect on the field vector E when the dir e c t o r is inclined. In c o n t r a s t t o this, in the problem
with a light field, a component E, a p p e a r s because of the
condition div D = 0.
e,,,,
As a r e s u l t , in equation (6) we have a n additional factor
the r o l e of which reduces to a c e r t a i n inc r e a s e of the threshold. Expanding (p(z, t ) in a F o u r i e r
s e r i e s and taking the boundary conditions ~ ( =z0 , t )
= cp(z = L, t ) = 0 into account, we obtain f r o m (6)
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Sov. Phys. JETP 54(1), July 1981
T h e e x a c t solution of this equation will be given in
Sec. 5 below. H e r e we s h a l l r e s t r i c t ourselves to only
taking into account t e r m s of o r d e r cp and (p3 in this
equation, a procedure valid n e a r the F r e d e r i c k s threshold. T h e s e a r c h f o r a solution of the f o r m
c p " f r sin ( n z i L )+cp, sin ( 3 n z l L )t
.. .,
which automatically s a t i s f i e s the boundary conditions,
leads to the following expression for (p,:
9 e.
cpt2=2 ( 1 ---4 ell
l,!
)
-' P-P ,,
-7q-l
-
H e r e (p, rp:. T h e r e f o r e rp, is proportional to the
s q u a r e r o o t of t h e e x c e s s of the intensity P above
threshold P, .
.
Zel'dovich eta/.
33
The advance of the optical phase #(L) over the cuvette
thickness L can be obtained from (8b); expanding (8b) to
terms -d we obtain
-' P-P F,
C
4. THE FREDERICKS TRANSITION I N NARROW
BEAMS
In the case where the transverse dimension of the
beam a, is smaller than o r on the order of the cuvette
thickness L, the Frank energy due to the transverse
gradients of the director becomes dominant. Let us
first estimate the order of magnitude of the threshold
power of the Fredericks transition. The energy of the
perturbed state is
Here 6 2 , ~is the volume occupied by the disturbance and
K is the Frank constant. This expression gives a stable
state cp, = O a t small IE 12, and instability s e t s in a t
As seen from expression (14), IE l2a; = const a t a,<< L.
In order to exactly determine the threshold of the
Fredericks transition, it is necessary to solve the
three-dimensional problem of the stability of the solution cp =O. We shall consider this problem for several
particular cases.
a. The single-constant approximation. In the singleconstant approximation we can assume even for a narrow beam that the vectors E and n lie in the xz plane.
) have
Then in the approximation linearized in ~ ( r we
Here P , ; is determined by formula (12) using the fact
that K l l =K.
With the substitution cp(r, t) = g'(r) e x p r t , equation
(15) becomes
of q for a "potential" of the form (18) will be1'
The eigenvalues of P from (17), corresponding to the
)
is P = n 2 / ~ ' . The perturbaeigenfunction ~ ( z=sin(nz/l)
tion of the director field cp'(r) will grow exponentially
in time at r = q - ' ~ ( q-P) > 0. Then, defining the threshold intensity of the Fredericks transition a s that value
of Poa t which a bound state first appears in the potential
well U a l/cosh2 px, we obtain
-
P th
(l+bL/n).
(20)
The quantity BL can be viewed a s the ratio of the cuvette
thickness L to the beam diameter d: PL- ~ / d .
b. Another case which can be solved analytically and
by which we can approximate the r e a l distribution of the
intensity in the beam of a light wave corresponds to a
function P(r) of the form
const for p G p ~
(2 1)
where p is the distance from the beam axis in the transverse direction. Writing equation (16) in cylindrical
coordinates and making the substitution cp'(r) = L(P)X(Z),
we find
The solution of (22a) is a zeroth-order Bessel function.
Let
P ( p C p o )> ( L l n ) ? h P ~=PF,
,
[in expression (23) we have s e t h =$/L', since directorfield perturbations which increase with time exist only
for r = q - l ~ ( h- g ) > 0, where g, just a s P in case (a),
equals I?/L'].
I t is easy to s e e that the condition (23) is
necessary for the existence of a nontrivial, that is, not
identically equal to zero, solution to the system of
Eqs. (22), compatible with the boundary conditions.
Then the solution of Eq. (22a) will be
S(P)=
Let us first consider the case where a "ribbon" light
beam falls on the medium, P(r,) =P(x). Then after
separating the variables in equation (16) we find by
means of the substitution cpl(r) =E(x)~(z)
(23)
(2 4)
for p>p0
where cl and c2 a r e constants and J, and K, a r e Bessel
functions. The condition of continuity of the logarithmic derivative at the point p = po gives
Equation (17a) is the one-dimensional Schrijdinger
equation for the potential U - -P(x). It is possible to
solve this equation analytically for a relation, for example, of the form
P ( z )= P ~ I C ~ ~ ~ ~ Z ,
(18)
that is, for a function which gives a good qualitative
approximation of a Gaussian (Fig. 2). The eigenvalues
34
Sov. Phys. JETP 54(1), July 1981
FIG. 2. Form of the function 1/cosh2x. The dashed line shows
the function exp(-x2).
Zel'dovich et a/.
34
=K,
(25)
(p)
/ KO(y).
In the case n p , / ~ < <1, if in addition we assume that
3(LL
<I,
(26)
PF~
we have from Eq. (25)
This occurs, for example, for the nematic crystal
PAA, for which &,= 0.9; K,, =4.5 x9.5 x lom7dyne [at
the temperature ~ = 1 2 5 " C(Ref. 13)] and B=-0.03 (B
=0.7 for MBBA and for B=0.06 for OCBP).
T o study the Fredericks transition for B s 0 we integrate Eq. (10) with respect t o z after multiplying it by
2dq/dz. After determining the integration constant in
t e r m s of the maximum angle of deviation of the director
from the unperturbed direction cp, we obtain
P th = P F[I-2LZlilZp.'
~
In (np,/L)
1.
(27)
As s e e n from expression (27), the condition (26) is
satisfied for ln(np0/L)>> -2, and this strengthens considerably the inequality np0/L<< 1.
In the case where p o 2 L, we obtain from Eq. (25)
JO,LK,(~P~~L)
npoK,(npdL)
.
1'1
dq
(di)
-
e,' P
~I-(e./e,l)sin~ql"'-[I-(~./ell)sin2q,l'~
rX.. ( I - ) sin'q) [I- (e./~,~)sin~q]"'[
1- ( ~ ~ / e ~ ~ ) s i n ~' ( o , , , " ~
(32)
Taking into account the fact that the maximum angle of
deviation cp, is reached a t the center of the cell, that
is, for z = ~ / 2 , we get from (32)
(28)
Ym
(1-k sinZq ) [I-(e./e,,)sin'q]'~[l- (en/~,)sin2
cp,]"
)" dq
Here Jo,=
2.4 is the f i r s t z e r o of the Bessel function
~ ~ ( 2 In
) . the limit po>>L formula (28) gives PI, =P,,, in
agreement with the result obtained in Sec. 3 for broad
beams.
j(
In the case where the beam radius is of the order of
the cuvette thickness, np,/L=: 1, Eq. (25) can be solved
graphically. A s a res~!lt we find
Assuming that the power density P of the radition
falling on a cell of the NLC is close to the threshold
value for the Fredericks transition, that i s , rp,c 1 , let
The
us compute the integral in (33) up to t e r m s -rp.:
solution of the resulting biquadratic equation for cp, has
the form
Therefore, in all the cases studied we have PI, -PI,
1/L2, whereas the coefficient of proportionality is
determined by the ratio p , / ~ of the beam radius to the
cuvette thickness.
-
In the case of more than a single constant the threshold
of the Fredericks transition can be determined analytically for a ribbon beam of the form P(r,) =P(y)polarized along the x axis. The equation linearized in ~ ( r in)
this case has the form
Introducing the new variable y' =(K,,/K~,)"'~, Eq. (30)
can be written in the form (17) by replacing x by y and
using the definition of P, from (12). After using the
solution to (17), we obtain for a light wave of the form
P(y) = Po/cosh2,9y
[ 1- (E,,/E,\)
si$ q ~ ' ~ - [ f ~- e ~ / e l l ) sq,,,]'h
ina
(33)
where
For the liquid crystals MBBA, OCBP, and PAA the
values of G a r e almost identical a t G;. 0.06. Before discussing formula (341, we note that it can in principle be
obtained from the condition of the minimum of the free
energy after expanding it in a s e r i e s in p, up to t e r m s
Apart from a coefficient that renders it dimensionless, the free energy has the form
At the point determined by'formula (34) we have
Therefore, when the beam dimensions a r e slightly
smaller than the curvette thickness the inclusion of
more than one constant leads to small corrections. We
can expect this to hold for other beam shapes, also.
5. HYSTERESIS OF THE FREDERICKS TRANSITIONS
I N BROAD BEAMS
Formula (11) becomes invalid for the maximum angle
of deviation of the director from the unperturbed direction if the parameters of the liquid-crystal medium a r e
such that
@&
IPm
FIG. 3. Dependence of the free energy iP on cp, for different
values of the light field power P: I-P 4 Pa,2 - P = P & , &Pth
. 4
35
Sov. Phys. JETP 54(1), July 1981
*P < Prr, -P > g,.
Zel'dovich eta/.
35
the field of the director in this case were discussed in
Secs. 3-5. Here we shall study the nature of the reorientation of the director by the field of an extraordinary
light wave in the intermediate region, that is, when the
wave propagates a t small angles to the director.
FIG. 4. Hysteresis of the Fredericks transition. The arrows
indicate the direction of variation of the power of the light
field P.
Substituting the expressions for the light fields (8a)
into Eq. (5) and including terms up to third order in the
small quantity rp = rp,sin(nz/~) (that is, assuming that
the power of the wave is less than o r of order P, :) and
of first order in s =sins, we find
Therefore, we should use the plus sign in formula (34)
(the condition that the free energy be a minimum).
-
If BaO the quantity cp, will be r e a l a t C = 1 ( P / P , , ) ' ~
S O , that is, P,, =P,,. If B< 0, then in order that rp, be
r e a l it is sufficient to require that the expression under
the square root in (34) be positive, B2- 4GC 3 0 , from
which we have
where sin y = &-In s i n a and y is the angle of refraction
of the wave in the cell of the NLC.
In the case of weak fields P<< P,, and large anisotropy
of the permittivity, Eq. (39) duplicates the results obtained earlier4:
cp,>,=e.IE IZLZ sin y/2n4K,,.
However, we s e e from (36) and (37) that when the power
density of the incident radiation is P =PI,, the quantity
cp, given by (34) is a point of inflection for the free
energy function (see Fig. 3). As seen from Fig. 3, for
values PI,< P < P,, the function 9(cp,) has a local minimum at cp,+ 0, where
The point p, = O becomes unstable only a t P>P,,.
From the above discussion it is clear that a s the power
density of the light wave increases from z e r o the Frede r i c k transition occurs at P=P,,, s o that a free-energy
minimum corresponding to cp#,
0 and occurring at PI,
< P < P,, is unattainable because of the presence of the
potential barrier. For the inverse transition, that is,
a s P decreases from the region of larger P, ., the freeenergy minimum reached a t P < P,, and corresponding to
p,# 0, while not an absolute minimum of the function 9
[as seen from (3811, is separated from it by the potential
barrier. The barrier only disappears a t P=P,, and then
we have rp, = 0 (Fig. 4).
6. PROPAGATION OF A LIGHT WAVE AT SMALL
ANGLES TO THE DIRECTOR
As already noted, an extraordinary wave propagating
a t an angle to the director induces strong nonlinear
optical effects a t very low powers P C P , , (Refs. 3-9).
When a light wave is normally incident on a cuvette with
homeotropic orientation of the NLC , however, reorientation of the director is possible only above some threshold value of the power of the wave. The distortions of
Since the formulas a r e quite awkward we shall not give
the analytic form of the solution of Eq. (39). The graph
of the function c p , ( ~ / ~,), is shown in Fig. 5 for the
parameters of the liquid crystal OCBP and for different
angles y.
The behavior of the cell in the oblique-incidence case
discussed here is analogous to the case of a cell in an
oblique magnetic field-in both cases there is no rigorous F r e d e r i c k transition (cf. Ref. 13, $4.2.3).
7. DISCUSSION
Let us estimate numerically the value of the threshold
power for the Fredericks transition for the nematic
crystal OCBP used in Ref. 11. The parameters of the
liquid crystal a r e & f a = 1.66, &i0= 1.53, Kll = 4 x
dyne, the curvette thickness is
dyne, K,, = 7 x
L = 150 p m , and the beam radius is p, = 50 pm. Substituting the Frank constant K =0.5 (K,, +K,,) into the
expression for P I,, we find from formula (29)
This value is in very good agreement with the experimental value"
For the cuvette thickness L =50 pm and the same values
of all the other parameters the value of the threshold
power estimated from (28) is
This value is 3.1 times larger than that for the Frederi c k transition in a cuvette of thickness L = 150 pm,
which is also in very good agreement with the results
of Ref. 11.
FIG. 5. Dependence of the maximum angle of deviation cp, on
P/P,, in the case of a light wave incident on a cell at small
angles: 1-y = l o 0 , Z-y= 5".
36
Sov. Phys. JETP 54(1), July 1981
The temperature dependence of the threshold power is
determined by the temperature dependence of the Frank
constant and of the permittivity. Bearing in mind the
fact that the Frank constant K varies with temperature
a s the square of the order parameter K-5? while the
anisotropy of the permittivity &,-S (Ref. 12), we can
Zel'dovich e t a / .
separate the temperature-dependent part in expression
(12) for P,, in the form PF,a c,E,~&;"~. AS the temperature'
increases, E, and d e c r e a s e s while E, increases; f r o m
this we s e e that a s the temperature increases the threshold
power decreases. Introducing the notation
we can write P,,
&
,, . Since &;I2 depends weakly on
the temperature (for example, in the region of the
MBBA mesophase &t12changes by only 0.01 of its value),
a s the point of the phase transition to a n isotropic
liquid is approached the threshold power decreases in
proportion to c,, that is, to the o r d e r parameter S.
Here we note the following. At s m a l l values of
and k, a s is the c a s e near the critical point, Eq. (33)
determiningthe maximum angle of deviation of the director cp, a s a function of the power of the light field can
be simplified. For this we must expand the integrand
in powers of
and k. Then the integral on the lefthand side of Eq. (33) is expressed in t e r m s of elliptic
integrals and the resulting equation implicitly determines the function cp, = cpm(P). In the limit where P/P, ,
<< 1 (that i s , (pm=n/2 6,6<< 1) the function cpm(P)can be
determined explicitly:
-
As was shown in Sec. 5, for certain types of liquid
crystals the Fredericks transition can be accompanied
by hysteresis. Let us calculate f o r the c a s e of the well
studied liquid crystal PAA the main characteristics of
this phenomenon. As s e e n from formula (37) with B
=-0.03 and G =0.06, the powers a t which the Fredericks
effect in PAA i s switched on and off differ little: P,,
= 0.9925 T h e angle corresponding to the appearance
of a local minimum of the free-energy function [that i s ,
pmcorresponding to the inflection point of the function
@(cpm)]is c p , ~ ( - ~ / 2 ~ )=0.5.
"~
,.
Let us demonstrate that thermal fluctuations of the
direction of the director do not lead to surmounting of
the potential b a r r i e r between the minima of the f r e e
energy function. F o r this we note that thermal fluctuations of the direction of the director would cause the
hysteresis t o disappear if the mean angle of the fluctuation deviations were on the order of o r larger than the
difference of the angles corresponding to the local minimum and maximum of the f r e e energy. T h i s difference
is easily calculated using formula (34) for some value
of the power P such that P,,.< P < P,,(that i s , O< C < B'/
40). For example, for C = B ' / ~ D we find A q =O.4, and
since ((6cp)2)"2<<1 for thermal fluctuations of the direct o r the local minimum of the f r e e energy CP.(cp,> 0) is
stable to thermal fluctuations.
In concluding this discussion we note that a Frederi c k ~transition induced by light fields in a NLC with
&,> 0 can occur only when the light wave is normally
incident on a homeotropically oriented NLC cell. I t is
impossible to have a Fredericks transition induced by
37
Sov. Phys. JETP 5411, July 1981
a n ordinary light wave normally incident on a planarly
oriented NLC cell. T h i s can be understood when it is
realized that the electric field of an ordinary light wave
in a NLC remains perpendicular to the director a s it
propagates through the medium. Strong nonlinear optical effects a r i s e if the polarization of the normally
incident cell makes s o m e angle with the director. We
have considered these effects in a n e a r l i e r study.'
T h e authors a r e grateful to E. I. Kats f o r a discussion
of these problems.
Note added inwoof. We have been kindly informed
by the authors of a paper submitted t o this journal that
they have also studied the theory of light-induced
F r e d e r i c k s transitions (A. S. Zolot'ko, V. F. Kitaeva,
N. N. Sobolev, and A. P. Sukhorukov, Zh. Eksp. Teor.
Fiz. 81, 933 (1981) [Sov. Phys. J E T P 54, No. 9 (1981)l)
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's. M. Arakelyan, G. L. Grigoryan, S. T s . Nersisyan, M. A.
Nshanyan, and Yu. S. Chilingaryan, ZhETF Pis. Red. 28,
202 (1978) [ J E T P Lett. 28, 186 (1979)l.
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Pilipetski, A. V. Sukhov, N. V. Tabiryan, and B. Ya,
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'9.
B. Pakhalov, A. S. Tumasyan, and Yu. S. Chilingaryan,
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Translated by Patricia Millard
Zel'dovich et at.